Multilevel Richardson-Romberg extrapolation

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1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

Abstract : We propose and analyze a Multilevel Richardson-Romberg $MLRR$ estimator which combines the higher order bias cancellation of the Multistep Richardson-Romberg $MSRR$ method introduced in Pages 07 and the variance control resulting from the stratification in the Multilevel Monte Carlo $MLMC$ method see Heinrich, 01 and Giles, 08. Thus we show that in standard frameworks like discretization schemes of diffusion processes an assigned quadratic error $\varepsilon$ can be obtained with our $MLRR$ estimator with a global complexity of $\log1-\varepsilon-\varepsilon^2$ instead of $\log1-\varepsilon^2-\varepsilon^2$ with the standard $MLMC$ method, at least when the weak error $\esp{Y h}-\esp{Y 0}$ of the biased implemented estimator $Y h$ can be expanded at any order in $h$. We analyze and compare these estimators on two numerical problems: the classical vanilla and exotic option pricing by Monte Carlo simulation and the less classical Nested Monte Carlo simulation.

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